On the geometrization of the Absolute Galois Group,--- the construction of Dessin d'Enfants

abstract

Belyi's theorem establishes an equivalence between the arithmetic property of a curve being defined on a number field and the geometric and analytic property of the existence of a meromorphic function with at most critical values. Grothendieck's idea in his Sketch of a Program was to utilize this idea to define a natural action of the Absolute Galois Group on families of graphs satisfying certain necessary properties on topological surfaces, and thereby develop a new tool for gaining a better understanding of this group. Subsequently, Drinfeld, Ihara and coworkers obtained injective homomorphisms from the Absolute Galois group into the group of outer automorphisms of the projective line minus three points and of the profinite completion of a sufficiently punctured Riemann sphere, and analyzed the image in the former case. One of the problems in this program has been the lack explicit examples linking the algebraic and geometric ideas. In this work a practical and explicit method is presented for constructing many examples and exhibiting the action of the Galois group in a geometric fashion on the relevant class of graphs which are the basic combinatorial/geometric objects.
(This is based on the work of my student Ali Kamalinejad)